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## Abstract

The solutions of Eady's 1949 model of baroclinic stability are extended numerically to include the non-geostrophic perturbations which wore not covered by the analysis in Part I. It is found that the largest growth rates are never associated with these new perturbations, so the tentative conclusions of Part I are verified. The more exact numerical solutions lead only to slight quantitative modifications of the results of Part I. If we let Ri be the Richardson number, then the largest growth rates are associated with “geostrophic” baroclinic instability if Ri>0.950; with symmetric instability if ¼<Ri<0.950; and with Kelvin-Helmholtz instability if 0<Ri<¼. Geostrophic baroclinic instability and symmetric instability can exist simultaneously if 0.84<Ri<1, and symmetric instability and Kelvin-Helmholtz instability can exist simultaneously if 0<Ri<¼

## Abstract

The solutions of Eady's 1949 model of baroclinic stability are extended numerically to include the non-geostrophic perturbations which wore not covered by the analysis in Part I. It is found that the largest growth rates are never associated with these new perturbations, so the tentative conclusions of Part I are verified. The more exact numerical solutions lead only to slight quantitative modifications of the results of Part I. If we let Ri be the Richardson number, then the largest growth rates are associated with “geostrophic” baroclinic instability if Ri>0.950; with symmetric instability if ¼<Ri<0.950; and with Kelvin-Helmholtz instability if 0<Ri<¼. Geostrophic baroclinic instability and symmetric instability can exist simultaneously if 0.84<Ri<1, and symmetric instability and Kelvin-Helmholtz instability can exist simultaneously if 0<Ri<¼

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## Abstract

The problem of the steady symmetric motion of a Boussinesq fluid is considered for a system with small aspect ratio. It is assumed that the motion is driven by applying a periodic heat flux to the horizontal boundaries. Solutions are first found for a non-rotating system in which nonlinear effects are small, but not zero. The solutions show that if the fluid is heated from above, the meridional circulation tends to be concentrated near the upper boundary at the point where the cooling is a maximum; when the fluid is heated from below the meridional circulation tends to be concentrated near the lower boundary at the paint where the heating is a maximum.

Then, it is shown for a non-rotating system that when nonlinear effects are dominant, vertical boundary layers must form. These vertical boundary layers form at points where the horizontal velocity is zero, and are characterized by small horizontal velocities and temperature gradients, but large vertical velocities and horizontal diffusion. By means of scaling analysis, the scales and magnitudes of the variables are determined for both the internal boundary layers and the boundary layers along the horizontal boundaries, when nonlinear effects are dominant.

Next, the effect of rotation is considered, and it shown that exactly the same sorts of vertical boundary layers will form in a rotating system. Scaling analysis is again used to show that in this case the horizontal boundary layers near the internal boundary layers are of the same kind as in the non-rotating case, but far enough away from the internal boundary layers they merge into a nonlinear Ekman layer.

Finally, some possible geophysical applications are considered. The model of the atmospheric circulations on Venus proposed by Goody and Robinson is found to agree qualitatively with the results presented here, but the quantitative results for the internal boundary layer, or mixing region, are found to differ considerably. Also, estimates are made for the internal boundary layer which would accompany a Hadley cell similar to that found in the earth's tropical region. It is found that the rising motions will occur over a region about 200 km in width. This result suggests that the nonlinear process which produces these internal boundary layers may be one of the important processes in determining the structure of the Intertropical Convergence Zone. Finally, the identification of the narrow sinking regions as another example of the kind of internal boundary layer studied here is considered, but in this case the magnitudes and scales are not plausible.

## Abstract

The problem of the steady symmetric motion of a Boussinesq fluid is considered for a system with small aspect ratio. It is assumed that the motion is driven by applying a periodic heat flux to the horizontal boundaries. Solutions are first found for a non-rotating system in which nonlinear effects are small, but not zero. The solutions show that if the fluid is heated from above, the meridional circulation tends to be concentrated near the upper boundary at the point where the cooling is a maximum; when the fluid is heated from below the meridional circulation tends to be concentrated near the lower boundary at the paint where the heating is a maximum.

Then, it is shown for a non-rotating system that when nonlinear effects are dominant, vertical boundary layers must form. These vertical boundary layers form at points where the horizontal velocity is zero, and are characterized by small horizontal velocities and temperature gradients, but large vertical velocities and horizontal diffusion. By means of scaling analysis, the scales and magnitudes of the variables are determined for both the internal boundary layers and the boundary layers along the horizontal boundaries, when nonlinear effects are dominant.

Next, the effect of rotation is considered, and it shown that exactly the same sorts of vertical boundary layers will form in a rotating system. Scaling analysis is again used to show that in this case the horizontal boundary layers near the internal boundary layers are of the same kind as in the non-rotating case, but far enough away from the internal boundary layers they merge into a nonlinear Ekman layer.

Finally, some possible geophysical applications are considered. The model of the atmospheric circulations on Venus proposed by Goody and Robinson is found to agree qualitatively with the results presented here, but the quantitative results for the internal boundary layer, or mixing region, are found to differ considerably. Also, estimates are made for the internal boundary layer which would accompany a Hadley cell similar to that found in the earth's tropical region. It is found that the rising motions will occur over a region about 200 km in width. This result suggests that the nonlinear process which produces these internal boundary layers may be one of the important processes in determining the structure of the Intertropical Convergence Zone. Finally, the identification of the narrow sinking regions as another example of the kind of internal boundary layer studied here is considered, but in this case the magnitudes and scales are not plausible.

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## Abstract

The meridional and vertical eddy fluxes of sensible heat produced by small-amplitude growing baroclinic waves are calculated using solutions to the two-level model with horizontal shear in the mean flow. The results show that the fluxes are primarily dependent on the local baroclinicity, i.e., the local value of the isentropic slopes in the mean state. Where the slope exceeds the critical value, the transports are poleward and upward; where the slope is less than the critical value, the transports are equatorward and downward.

These results are used to improve an earlier parameterization of the tropospheric eddy fluxes of sensible heat based on Eady's model. Comparisons with observations show that the improved parameterization reproduces the observed magnitude and sign of the eddy fluxes and their vertical variations and seasonal changes, but the maximum in the poleward flux is too near the equator. The corresponding parameterizations for the eddy coefficients describing the transport of any conserved quantity are given.

## Abstract

The meridional and vertical eddy fluxes of sensible heat produced by small-amplitude growing baroclinic waves are calculated using solutions to the two-level model with horizontal shear in the mean flow. The results show that the fluxes are primarily dependent on the local baroclinicity, i.e., the local value of the isentropic slopes in the mean state. Where the slope exceeds the critical value, the transports are poleward and upward; where the slope is less than the critical value, the transports are equatorward and downward.

These results are used to improve an earlier parameterization of the tropospheric eddy fluxes of sensible heat based on Eady's model. Comparisons with observations show that the improved parameterization reproduces the observed magnitude and sign of the eddy fluxes and their vertical variations and seasonal changes, but the maximum in the poleward flux is too near the equator. The corresponding parameterizations for the eddy coefficients describing the transport of any conserved quantity are given.

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## Abstract

A parameterization for the fluxes of sensible heat by large-scale eddies developed in an earlier paper is incorporated into a model for the mean temperature structure of an atmosphere including only these fluxes and the radiative fluxes. The climatic changes in this simple model are then studied in order to assess the strength of the dynamical feedback and to gain insight into how dynamical parameters may change in more sophisticated climatic models. The model shows the following qualitative changes: 1) an increase in the solar constant leads to increased static stability, decreased dynamic stability, and stronger horizontal and vertical winds; 2) an increase in the amount of atmospheric absorption leads to decreased static and dynamic stability, and stronger horizontal and vertical winds; and 3) an increase in rotation rate leads to greater static and dynamic stability, weaker horizontal winds, and stronger vertical winds. The quantitative results provide support for the common assumption that the static stability remains constant during climatic changes. Twenty-five percent changes in the external parameters cause changes in the static stability of the order of only a few tenths of a degree per kilometer. The results also show that the assumption that the horizontal eddy flux can be represented by a diffusion law with a constant eddy coefficient is a bad one, because of the strong negative feedback in the eddy fluxes.

## Abstract

A parameterization for the fluxes of sensible heat by large-scale eddies developed in an earlier paper is incorporated into a model for the mean temperature structure of an atmosphere including only these fluxes and the radiative fluxes. The climatic changes in this simple model are then studied in order to assess the strength of the dynamical feedback and to gain insight into how dynamical parameters may change in more sophisticated climatic models. The model shows the following qualitative changes: 1) an increase in the solar constant leads to increased static stability, decreased dynamic stability, and stronger horizontal and vertical winds; 2) an increase in the amount of atmospheric absorption leads to decreased static and dynamic stability, and stronger horizontal and vertical winds; and 3) an increase in rotation rate leads to greater static and dynamic stability, weaker horizontal winds, and stronger vertical winds. The quantitative results provide support for the common assumption that the static stability remains constant during climatic changes. Twenty-five percent changes in the external parameters cause changes in the static stability of the order of only a few tenths of a degree per kilometer. The results also show that the assumption that the horizontal eddy flux can be represented by a diffusion law with a constant eddy coefficient is a bad one, because of the strong negative feedback in the eddy fluxes.

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## Abstract

It is suggested that the apparent lag of Jupiter's mean rotation rate in extratropical latitudes (System II) behind the rotation rate of Jupiter's radio emissions (System III) is caused by the difference between phase speeds and true speeds in extratropical latitudes. An estimate of the difference based on the formula for the phase speed of Rossby waves agrees with the difference calculated from the two rotation rates.

## Abstract

It is suggested that the apparent lag of Jupiter's mean rotation rate in extratropical latitudes (System II) behind the rotation rate of Jupiter's radio emissions (System III) is caused by the difference between phase speeds and true speeds in extratropical latitudes. An estimate of the difference based on the formula for the phase speed of Rossby waves agrees with the difference calculated from the two rotation rates.

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## Abstract

A simple model for the structure of a non-rotating Hadley regime in an atmosphere with large thermal inertia is developed. The radiative fluxes are estimated by using a linearization about the radiative equilibrium state and the dynamical fluxes are estimated by using scaling analysis. The requirement that differential heating by these fluxes be in balance in both the meridional and vertical directions leads to two equations for the mean static stability and meridional temperature contrast. The solution depends on two parameters: the strength of the radiative heating, as measured by the static stability *A _{e}
* of the radiative equilibrium state; and the ratio of the time it takes an external gravity wave to traverse the atmosphere to the time it would take the atmosphere to cool off radiatively, denoted by ε.

In the deep Venus atmosphere ε ≈ 10^{−5}; the equations are therefore analyzed in the limit ε → 0. The large-scale dynamics has virtually the same effect on the lapse rate as small-scale convection: if *A _{e}
* > 0 the radiative lapse rate is unchanged, while if

*A*< 0 the lapse rate becomes subadiabatic, but only by an amount of order ε

_{e}^{⅔}. Therefore, one need not invoke convection to explain the approximate adiabatic lapse rate in the Venus atmosphere, but a greenhouse effect is necessary to explain the high surface temperatures. The other properties of the solutions when

*A*< 0 are consistent with observational evidence for the deep atmosphere: the horizontal velocities are typically ∼2 m sec

_{e}^{−1}, the vertical velocities ∼½ cm sec

^{−1}, and the meridional temperature contrast is unlikely to exceed 0.1K.

The same approach is used to study the time-dependent problem and determine how long it would take for a perturbed atmosphere to reach equilibrium. If *A _{e}
* > 0 the adjustment is primarily governed by the radiative time scale, which is about 100 earth years for the deep Venus atmosphere. If

*A*< 0 the adjustment is governed by an advective time scale which may be as short as 20 earth days. Published numerical studies of the deep circulation have only treated the first case, but their integrations were not carried beyond about 200 earth days and therefore do not describe true equilibrium states. Only the second case,

_{e}*A*< 0, is consistent with the observations and it would be relatively easy to study numerically.

_{e}## Abstract

A simple model for the structure of a non-rotating Hadley regime in an atmosphere with large thermal inertia is developed. The radiative fluxes are estimated by using a linearization about the radiative equilibrium state and the dynamical fluxes are estimated by using scaling analysis. The requirement that differential heating by these fluxes be in balance in both the meridional and vertical directions leads to two equations for the mean static stability and meridional temperature contrast. The solution depends on two parameters: the strength of the radiative heating, as measured by the static stability *A _{e}
* of the radiative equilibrium state; and the ratio of the time it takes an external gravity wave to traverse the atmosphere to the time it would take the atmosphere to cool off radiatively, denoted by ε.

In the deep Venus atmosphere ε ≈ 10^{−5}; the equations are therefore analyzed in the limit ε → 0. The large-scale dynamics has virtually the same effect on the lapse rate as small-scale convection: if *A _{e}
* > 0 the radiative lapse rate is unchanged, while if

*A*< 0 the lapse rate becomes subadiabatic, but only by an amount of order ε

_{e}^{⅔}. Therefore, one need not invoke convection to explain the approximate adiabatic lapse rate in the Venus atmosphere, but a greenhouse effect is necessary to explain the high surface temperatures. The other properties of the solutions when

*A*< 0 are consistent with observational evidence for the deep atmosphere: the horizontal velocities are typically ∼2 m sec

_{e}^{−1}, the vertical velocities ∼½ cm sec

^{−1}, and the meridional temperature contrast is unlikely to exceed 0.1K.

The same approach is used to study the time-dependent problem and determine how long it would take for a perturbed atmosphere to reach equilibrium. If *A _{e}
* > 0 the adjustment is primarily governed by the radiative time scale, which is about 100 earth years for the deep Venus atmosphere. If

*A*< 0 the adjustment is governed by an advective time scale which may be as short as 20 earth days. Published numerical studies of the deep circulation have only treated the first case, but their integrations were not carried beyond about 200 earth days and therefore do not describe true equilibrium states. Only the second case,

_{e}*A*< 0, is consistent with the observations and it would be relatively easy to study numerically.

_{e}^{ }

## Abstract

In order to obtain estimates of the static stability in rotating atmospheres without performing numerical integrations of the equations of motion, a simple model is developed in which the radiative flux of heat is assumed to be balanced by the fluxes of sensible beat and potential energy due to large-scale eddies. The radiative flux divergence is modeled by a linearization about the radiative equilibrium state and the dynamical fluxes are modeled by calculating correlations from stability theory and by assuming that the amplitudes are limited by nonlinear effects. From the energy equation a single algebraic equation is derived for the mean equilibrium value of the Richardson number, Ri, in the troposphere. The radiative equilibrium state is assumed to be known. Once the solution for RI is found, the mean vertical and meridional gradients of potential temperature, 〈∂θ/∂*z*〉 and 〈∂θ/∂*y*〉, and the main properties of the mean zonal wind and eddies can be easily calculated. Even though many important fluxes are left out of the model, it is capable of giving good qualitative results because of the strong feedback in the dynamical fluxes.

When applied to the earth, the model yields the values 〈∂θ/∂*z*〉≈+2K km^{−1}, 〈∂θ/∂*y*〉≈−0.4K (100 km)^{−1}, Ri≈30. These values are much more realistic than can he obtained from the traditional assumption of radiative-convective equilibrium, although the static stability is still about one-half the observed value because of neglected fluxes. When applied to Mars the model yields the values 〈∂θ/∂*z*〉≈+2K km^{−1}, 〈∂θ/∂*y*〉≈−1.2K (100 km)^{−1}, Ri≈10. which are in good agreement with the Mariner observations and with the numerical results of Leovy and Mintz. The destabilization of the Martian atmosphere compared to the earth's (i.e., the smaller value of Ri) is due to the much shorter radiative relaxation time on Mars, which makes the radiative fluxes more efficient at destabilizing the atmosphere. When applied to Jupiter the model yields the values 〈∂θ/∂*z*〉≈10^{−4}K km^{−1}, 〈∂θ/∂*y*〉≈−2K (10,000 km)^{−1}. The destabilization of Jupiter's atmosphere compared to Mars’ and the earth's is caused by the small horizontal temperature gradients, which in turn are due to both the large scale of Jupiter and the presence of an internal heat source. These small gradients lead to relatively weak large-scale motions, so that the dynamical fluxes are less efficient at stabilizing the atmosphere. The value of Ri found for Jupiter is subject to large error because of its sensitivity to the values of the external parameters. It may lie anywhere in the range O<Ri<20. Consequently, Jupiter's dynamical regime cannot be specified with certainty, but the results do show that Jupiter's mean temperature structure will in any case be very near radiative-convective equilibrium.

The model is also used to study how an atmosphere will adjust to deviations from equilibrium. In the case of the earth and Mars the deviations from equilibrium trace out damped oscillations, while in the case of Jupiter they are simply damped. The *c*-folding time for the damping is 34 days for the earth, 3 days for Mars, and 16 years for Jupiter. The adjustment time in all three cases is primarily determined by the radiative relaxation time.

## Abstract

In order to obtain estimates of the static stability in rotating atmospheres without performing numerical integrations of the equations of motion, a simple model is developed in which the radiative flux of heat is assumed to be balanced by the fluxes of sensible beat and potential energy due to large-scale eddies. The radiative flux divergence is modeled by a linearization about the radiative equilibrium state and the dynamical fluxes are modeled by calculating correlations from stability theory and by assuming that the amplitudes are limited by nonlinear effects. From the energy equation a single algebraic equation is derived for the mean equilibrium value of the Richardson number, Ri, in the troposphere. The radiative equilibrium state is assumed to be known. Once the solution for RI is found, the mean vertical and meridional gradients of potential temperature, 〈∂θ/∂*z*〉 and 〈∂θ/∂*y*〉, and the main properties of the mean zonal wind and eddies can be easily calculated. Even though many important fluxes are left out of the model, it is capable of giving good qualitative results because of the strong feedback in the dynamical fluxes.

When applied to the earth, the model yields the values 〈∂θ/∂*z*〉≈+2K km^{−1}, 〈∂θ/∂*y*〉≈−0.4K (100 km)^{−1}, Ri≈30. These values are much more realistic than can he obtained from the traditional assumption of radiative-convective equilibrium, although the static stability is still about one-half the observed value because of neglected fluxes. When applied to Mars the model yields the values 〈∂θ/∂*z*〉≈+2K km^{−1}, 〈∂θ/∂*y*〉≈−1.2K (100 km)^{−1}, Ri≈10. which are in good agreement with the Mariner observations and with the numerical results of Leovy and Mintz. The destabilization of the Martian atmosphere compared to the earth's (i.e., the smaller value of Ri) is due to the much shorter radiative relaxation time on Mars, which makes the radiative fluxes more efficient at destabilizing the atmosphere. When applied to Jupiter the model yields the values 〈∂θ/∂*z*〉≈10^{−4}K km^{−1}, 〈∂θ/∂*y*〉≈−2K (10,000 km)^{−1}. The destabilization of Jupiter's atmosphere compared to Mars’ and the earth's is caused by the small horizontal temperature gradients, which in turn are due to both the large scale of Jupiter and the presence of an internal heat source. These small gradients lead to relatively weak large-scale motions, so that the dynamical fluxes are less efficient at stabilizing the atmosphere. The value of Ri found for Jupiter is subject to large error because of its sensitivity to the values of the external parameters. It may lie anywhere in the range O<Ri<20. Consequently, Jupiter's dynamical regime cannot be specified with certainty, but the results do show that Jupiter's mean temperature structure will in any case be very near radiative-convective equilibrium.

The model is also used to study how an atmosphere will adjust to deviations from equilibrium. In the case of the earth and Mars the deviations from equilibrium trace out damped oscillations, while in the case of Jupiter they are simply damped. The *c*-folding time for the damping is 34 days for the earth, 3 days for Mars, and 16 years for Jupiter. The adjustment time in all three cases is primarily determined by the radiative relaxation time.

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## Abstract

The results of Parts I and II are used to calculate the transports of heat and momentum that accompany growing baroclinic instabilities in Eady's model. The transports are calculated for both the conventional (“geostrophic”) kind of baroclinic instability and for symmetric instability, without any restriction on the stratification, as measured by the Richardson number. The transports are calculated consistently to second order in the amplitude expansion of stability theory, so that the transports are the sum of an eddy transport term and a mean transport term.

The results show that both kinds of instability always transport heat upward and poleward, and always transport zonal momentum downward. Under geostrophic conditions the horizontal transport of zonal momentum depends on the horizontal shear of the basic flow. This shear is neglected in Eady's model so the horizontal momentum transports calculated here only contain the non-geostrophic contribution to the transport. The results show that this non-geostrophic transport is always equatorward for geostrophic instability, but for symmetric instability it may be either equatorward or poleward depending on the value of the Richardson number. It is suggested that the equatorward transport of zonal momentum by geostrophic instability is a more likely mechanism for Jupiter's equatorial acceleration than the transport by symmetric instability.

## Abstract

The results of Parts I and II are used to calculate the transports of heat and momentum that accompany growing baroclinic instabilities in Eady's model. The transports are calculated for both the conventional (“geostrophic”) kind of baroclinic instability and for symmetric instability, without any restriction on the stratification, as measured by the Richardson number. The transports are calculated consistently to second order in the amplitude expansion of stability theory, so that the transports are the sum of an eddy transport term and a mean transport term.

The results show that both kinds of instability always transport heat upward and poleward, and always transport zonal momentum downward. Under geostrophic conditions the horizontal transport of zonal momentum depends on the horizontal shear of the basic flow. This shear is neglected in Eady's model so the horizontal momentum transports calculated here only contain the non-geostrophic contribution to the transport. The results show that this non-geostrophic transport is always equatorward for geostrophic instability, but for symmetric instability it may be either equatorward or poleward depending on the value of the Richardson number. It is suggested that the equatorward transport of zonal momentum by geostrophic instability is a more likely mechanism for Jupiter's equatorial acceleration than the transport by symmetric instability.

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